I am a student completing A levels and recently received the following question in an official mock exam for the Edexcel A-level in Mathematics. Having discussed the question extensively with my peers and teachers but to no avail, I would appreciate if if you could shed some light on where I went wrong.
The question:
An ecologist is investigating the number of lions and wildebeest in a national park.
The number of lions, measured in hundreds, $L$, is modelled by the equation $L = 1 + 2e^{0.03t}$ where $t$ is the number of years from the start of the investigation.
The number of wildebeest, measured in thousands, $W$ is modelled by the equation $W = 10 + 4e^{-0.03t}$ where $t$ is the number of years from the start of the investigation.
When $t = T$, according to the models, there are 10 times as many lions as wildebeest. Find the value of $T$ to one decimal place.
My solution:
Since $L$ is measured in the hundreds, I defined a new variable $L'$ as the actual number of lions. Since $L$ is measured in the hundreds, I found that $L' = 100L$. Similarly for wildebeest, I defined $W'$ as the actual number of wildebeest. Since $W$ is measured in the thousands, I found that $W' = 1000W$.
If when $t = T$ the number of lions is 10 times the number of wildebeest, that means $L' = 10W'$. Substituting the values for $L'$ and $W'$, this means $100L = 10000W$ and so $L = 100W$.
Solving the equations when this condition is true gave the following:
$1 + 2e^{0.03T} = 100(10 + 4e^{-0.03T}) \Rightarrow 2e^{0.06T} - 999e^{0.03T} - 400 = 0$
$\require{cancel} e^{0.03T} \approx 499.9, \cancel{-0.4} \Rightarrow 0.03T \approx 6.21 \Rightarrow T \approx 207.1$ years
Mark scheme solution (word for word):
Set $L = W$
$1 + 2e^{0.03T} = 10 + 4e^{-0.03T} \Rightarrow 2(e^{0.03T})^2 - 9 - 4e^{0.03T} = 0$
$e^{0.03T} \approx 4.907, \cancel{-0.4075} \Rightarrow 0.03T \approx ln(4.907) \Rightarrow T \approx 53.0$ years
Final:
When I substituted $t = 207.1$ years into the expressions for $L$ and $W$, I got the values 100080 and 10.008 respectively which seem to satisfy the conditions considering $L$ is in hundreds and $W$ is in the thousands.
Thank you for taking a look at this, I'd appreciate any advice you may have about this. As you can imagine, this was an exam question and so I really would like to clarify this.